We provide a comparative treatment of some aspects of spectral theory for self-adjoint and non-self-adjoint (but J-self-adjoint) Dirac-type operators connected with the defocusing and focusing nonlinear Schrödinger equation, of relevance to nonlinear optics. In addition to a study of Dirac and Hamiltonian systems, we also introduce the concept of Weyl-Titchmarsh half-line m-coefficients (and 2 × 2 matrix-valued M-matrices) in the non-self-adjoint context and derive some of their basic properties. We conclude with an illustrative example showing that crossing spectral arcs in the non-self-adjoint context imply the blowup of the norm of spectral projections in the limit where the crossing point is approached.
S. L. Clark and F. Gesztesy, "On Self-Adjoint and J-Self-Adjoint Dirac-Type Operators: A Case Study," Contemporary Mathematics, American Mathematical Society, Jan 2006.
Mathematics and Statistics
National Science Foundation (U.S.)
Keywords and Phrases
Dirac Operators; J-Self-Adjointness; Spectral Theory
Article - Journal
© 2006 American Mathematical Society, All rights reserved.